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Will an observer on the ground hear a sonic boom if a plane passes overhead at exactly the speed of sound? That is, the plane does not cross the sound barrier. The plane just hits Mach one - and continues to travel at that exact speed.

This is a bit of a problem because the speed of sound changes with altitude, so we assume it remains at the exact same altitude. I imagine there are other caveats also!

I would guess that there is a sonic boom, but I'm not sure.

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    $\begingroup$ Another caveat people often miss: if the plane flies at Mach 0.999, but the airflow over the wing is faster than the incident flow... is the plane technically supersonic? $\endgroup$ – AEhere Aug 11 '17 at 8:51
  • $\begingroup$ @AEhere But for the sound energy to concentrate in a shock wave, you need to consider the environment between the plane and the observer. Things that happen very close to the plane are not interesting. Actually, all jet planes would be technically supersonic by this consideration as their fan is supersonic :-) $\endgroup$ – yo' Aug 11 '17 at 10:02
  • $\begingroup$ Yes but the speed of sound for that specific environment: temperature, humidity, density, etc $\endgroup$ – jean Aug 11 '17 at 11:39
  • $\begingroup$ @AEhere When flow in some areas is supersonic and in other areas is subsonic it is called transsonic $\endgroup$ – TomMcW Aug 11 '17 at 18:15
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No, if standard atmospheric conditions apply.

Since speed of sound is proportional to temperature and temperature normally decreases with altitude, the speed of the aircraft at Mach 1 is subsonic in warmer, lower air. This means also that the Mach shock will diffuse and be audible on the ground either as a protracted rumble or not at all, depending on the aircraft's altitude.

Note that I assumed that the listener is located below the flight path of the aircraft. If that would not be the case, the noise of the aircraft diving below the listener altitude would be substantial.

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Not only does it produce a sonic boom at Mach 1, it will be the loudest boom that that particular aircraft can produce. enter image description here Image source

In the picture, 4 is the shok wave. The one at M1 is vertical, and has all the concentrated pressure differential in it. An aircraft flying at supersonic speeds forms two shock cones, one at the nose and one at the tail, with a pressure distribution in between that spreads the pressure differential out.

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  • $\begingroup$ But the total power contained in the shockwave must surely be higher at higher speed? Power is force times velocity, so I would imagine that the power dissipated by the shockwave grows quadratically with speed. $\endgroup$ – Sanchises Aug 11 '17 at 9:17
  • $\begingroup$ @Sanchises "power is force times velocity" -- why should it be so also for the vibrations the plane induces on the surrounding environment? (I don't say it isn't, I just don't see the reason.) $\endgroup$ – yo' Aug 11 '17 at 9:59
  • $\begingroup$ @Sanchises the output of the sonic boom itself is a step input in pressure at M=1, a more gradual pressure increase to the same end value when speed is higher. $\endgroup$ – Koyovis Aug 11 '17 at 10:15
  • $\begingroup$ @Sanchises the catch is in the figure .2 you see many shock waves "merging"/"adding" in a single big one boom $\endgroup$ – jean Aug 11 '17 at 13:04
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    $\begingroup$ @Koyovis I do believe you, but I think the explanation is a bit lacking. I can see that the shock is spread out over a cone rather than a flat disk (so I suppose it scales with $\tan\alpha$ or something), but how does the intensity of the shockwave come in to play, if at all? $\endgroup$ – Sanchises Aug 11 '17 at 13:45

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